Discrete Time Frequency Domain Plot for Sampled Signal x(t) = 3cos(1000πt+ 0.8π)

Let's start by finding the discrete time signal x[n] by sampling x(t) at a given sampling frequency fs.

The sampling frequency fs is the number of samples taken per second. In other words, it determines how often we sample the continuous time signal x(t).

The relationship between the continuous time signal x(t) and the discrete time signal x[n] is given by:

x[n] = x(n*T) where T is the sampling period, which is the inverse of the sampling frequency: T = 1/fs.

In this case, we are given x(t) = 3cos(1000πt+ 0.8π), so we need to find x[n] by sampling x(t) at a given sampling frequency fs.

Let's assume the sampling frequency fs = 5000 Hz.

The sampling period T = 1/fs = 1/5000 = 0.0002 seconds.

To find x[n], we need to evaluate x(t) at discrete time instances t = n*T.

x[n] = 3cos(1000π(nT)+ 0.8π) = 3cos(1000πn0.0002+ 0.8π) = 3cos(0.2πn+ 0.8π)

Now that we have x[n], we can find its discrete time frequency domain plot by taking the discrete Fourier transform (DFT) of x[n].

Let's assume we take N = 1024 samples of x[n].

The discrete Fourier transform (DFT) of x[n] is given by:

X[k] = Σ(x[n] * exp(-j2πk*n/N)), for n = 0 to N-1

where X[k] represents the frequency domain representation of x[n].

Let's calculate X[k] for k = 0 to N-1:

X[k] = Σ(3cos(0.2πn+ 0.8π) * exp(-j2πk*n/1024)), for n = 0 to 1023

We can use a software tool like MATLAB or Python to calculate X[k] for us.

Once we have X[k], we can plot the magnitude spectrum of X[k] to visualize the frequency domain representation of x[n].